What could possibly go wrong if Zerodha or Groww goes bankrupt and must shut down?

The broker is merely a broker. The broking company provides you with a platform to transact, and you remain the owner.
A Demat account is where your bought/sold stocks are maintained. CDSL/NSDL are centralized agencies under which Demat accounts are opened by stock brokers.

On a slightly humorous note, what happens if you rent your house to someone through a broker, and the broker goes bankrupt?

To end customers, momentary mental agony will happen.

No loss of shares and any other financial assets will be there.

Shares, mutual fund units, bond units etc are not with these brokers. So that is not going to be affected. The funds of clients held by these brokers is supposed to be in a separate account, but if there is a lapse in corporate governance, it can be affected

If Zerodha goes down either management f*cked up or the government.

Besides the shares in demat, they can not use the money you have kept with them. Everytime you add money, your balance is updated with exchange. When you withdraw it is again notified to exchange before it is credited to you. That is why the instant withdrawal facility they have only works during the day. After market hours exchange is closed. So they can alert exchange only next day (around 8am) when the go ahead is received and they trigger credit to your account.
You would also be getting monthly disclosure about your balance from cdsl/exchange. If no transaction in 30 days, all the money is sent back to you. Mandatory settlement at 90days (first friday of every quarter). You will be returned all the money youbhave with the broker.

If no transaction in the demat account for 365 days then it is marked dormant by the exchange. Broker can't touch your shares even if you had given a PoA. You need to do KYC.

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